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We study the potential speed of an optical neural network that uses the longitudinal cavity modes of an external-cavity laser diode as neurons. For this purpose, we used a laser diode coupled to two external cavities, each corresponding to one longitudinal cavity mode. The process of longitudinal mode switching is investigated for the case of intracavity optical modulation. In this experiment, the feedback for the mode in one cavity is modulated, and the length of the other cavity can be controlled. Three limitations are imposed on the switching speed. A number of external-cavity round trips are needed to switch from one mode to the other. It is observed that, depending on the amount of optical feedback in both cavities, between 7 and 21 round trips are needed. When the experimental results for varying cavity length are extrapolated to zero cavity length, a residual delay of a few nanoseconds remains. It is believed that this delay is due to a change in carrier density, needed to switch from one mode to another. Modified rate equations are used to model our experiments. The results of numerical simulations are in good agreement with the experimental results and predict the residual delay. The model also predicts a turn-on delay that is related to relaxation oscillations and imposes a third limitation on the operation speed of our optical neural network. Implications of our findings on the potential operation speed of the optical neural network are discussed and suggestions are made for optimization.